Superconvergence structures for rectangular and triangular finite elements are summarized. Two debatable issues in Zhu's paper [18] are discussed. A superclose polynomial to cubic triangular finite element is cons...Superconvergence structures for rectangular and triangular finite elements are summarized. Two debatable issues in Zhu's paper [18] are discussed. A superclose polynomial to cubic triangular finite element is constructed by area coordinate.展开更多
A new structure of superconvergence for the cubic triangular finite element approximation u/2 to a second-order elliptic problem Au=f is studied based on some orthogonal expansions in an interval. Suppose that Ω is a...A new structure of superconvergence for the cubic triangular finite element approximation u/2 to a second-order elliptic problem Au=f is studied based on some orthogonal expansions in an interval. Suppose that Ω is a convex polygonal domain with boundary L, its triangulation is uniform and T<sub>h</sub> is a set of vertexes and side midpoints of all elements. Then u<sub>h</sub> itself has no superconvergence points in Ω, while in any interior subdomain Ω<sub>0</sub> the average gradient Du<sub>h</sub> has superconvergence D（d<sub>h</sub>-u）=O(h<sup>m+1</sup>lnh) at z∈T<sub>h</sub>∩Ω<sub>0</sub>（no other superconvergence points）. Furthermore, prescribe u=0 on L<sub>1</sub>.Then the superconvergence near L<sub>1</sub> will surely disappear; if αa<sub>v</sub>u+bu=0 on L<sub>3</sub>, where v is the conormal direction, the numercal experiments show superconvergence up to L<sub>3</sub>（the case of A=-Δ and b=0 has already been proved）.展开更多
Based on an orthogonal expansion and orthogonality correction in an element, superconvergenceat symmetric points for any degree rectangular serendipity finite element approximation to second order ellipticproblem is p...Based on an orthogonal expansion and orthogonality correction in an element, superconvergenceat symmetric points for any degree rectangular serendipity finite element approximation to second order ellipticproblem is proved, and its behaviour up to the boundary is also discussed.展开更多
基金This work was supported by The Special funds of State Major Basic Research Projection (No. G1999032804) The National Natural Science Foundation of China (No. 19331021).
文摘Superconvergence structures for rectangular and triangular finite elements are summarized. Two debatable issues in Zhu's paper [18] are discussed. A superclose polynomial to cubic triangular finite element is constructed by area coordinate.
文摘A new structure of superconvergence for the cubic triangular finite element approximation u/2 to a second-order elliptic problem Au=f is studied based on some orthogonal expansions in an interval. Suppose that Ω is a convex polygonal domain with boundary L, its triangulation is uniform and T<sub>h</sub> is a set of vertexes and side midpoints of all elements. Then u<sub>h</sub> itself has no superconvergence points in Ω, while in any interior subdomain Ω<sub>0</sub> the average gradient Du<sub>h</sub> has superconvergence D（d<sub>h</sub>-u）=O(h<sup>m+1</sup>lnh) at z∈T<sub>h</sub>∩Ω<sub>0</sub>（no other superconvergence points）. Furthermore, prescribe u=0 on L<sub>1</sub>.Then the superconvergence near L<sub>1</sub> will surely disappear; if αa<sub>v</sub>u+bu=0 on L<sub>3</sub>, where v is the conormal direction, the numercal experiments show superconvergence up to L<sub>3</sub>（the case of A=-Δ and b=0 has already been proved）.
基金This work was supported by the Special Funds of State Major Basic Research Projects (Grant No. G1999032804) the National Natural Science Foundation of China (Grant No. 19871027).
文摘Based on an orthogonal expansion and orthogonality correction in an element, superconvergenceat symmetric points for any degree rectangular serendipity finite element approximation to second order ellipticproblem is proved, and its behaviour up to the boundary is also discussed.
